House Speaker John Boehner (L) arrives at a meeting of Republican members of the House of Representatives at the US Capitol on September 28, 2013 in Washington.(NICHOLAS KAMM/AFP/Getty Images)

One of the few paths to averting a government shutdown before midnight tonight is for House Speaker John Boehner to give up the fight to delay the new federal health law. If that happened, Mr. Boehner would need to make the case to his fellow House Republicans that they should end their standoff over the law, at least for now, with the Senate.

Signs are already appearing of what the GOP message might be in that case: The battle against the health law has obscured a bigger fight over spending—which Republicans are winning.

“This is a huge victory that nobody has talked about,” Rep. Mario Diaz-Balart (R., Fla.), a member of the House Appropriations Committee, said in an interview Sunday. “The spending portion of the spending bill we have won.”

That argument might help persuade Republicans to vote today to fund federal agencies, even if the measure includes none of the limits on the health law they’ve been pushing for.

The short-term bill at the center of the debate would keep the government funded for the next few weeks at current spending levels, an annual rate of $986 billion over the year. After that, lawmakers will have to extend funding again.

But unless Congress sets a different course, across-the-board cuts known as the sequester will be triggered in January, bringing spending down to $967 billion.

Congress hasn’t laid out ways yet to make those additional spending cuts, and doing so is already proving tricky even in the Republican-led House, which favors the cuts. When the House tried to pass a transportation and housing bill earlier this year that adhered to the lower spending number, GOP leaders had trouble rounding up votes. They wound up pulling the legislation from the floor.

Senate Democrats want to spend more than the $967 billion level that would be set under the sequester; they’ve been planning spending bills that add up to $1.058 trillion. And they plan to fight in the future to replace the sequester or to diminish its cuts.

But for now, the short-term spending bill now before Congress—the one that could avert the shutdown—does nothing to change those scheduled budget cuts.

GOP lawmakers such as Mr. Diaz-Balart view the short-term spending bill as a lopsided victory on the fiscal front.

“They literally took our number,’’ he said of the Senate. “It wasn’t halfway.’’

Other Republicans are also giving a preview of arguments aimed at persuading GOP lawmakers not to push the health law fight so hard now that it prompts a government shutdown. Some argued Sunday that changes to the health-care law could be exacted outside of the budget fight.

Rep. Charles Boustany (R., La.) noted that Republicans have already had success convincing Democrats to do away with pieces of the health-care law, such as the so-called 1099 provision, a tax-reporting requirement affecting small-business owners.

“It took time to get Democratic support,” Mr. Boustany said in an interview. “There is a smart way to do this that actually gets results. Going into a shutdown is going to have detrimental effects on the economy. I think Republicans are going to get blamed, and it really doesn’t do a whole lot to achieve the long-term goal” of dismantling the health-care law.

Some Democrats and President Barack Obama backed a repeal of the 1099 measure, saying it showed they were willing to work with Republicans to improve the health-care law.

For their part, Democrats have pushed to shorten the length of the stopgap spending bill. The Senate wants the measure to fund the government through mid-November, a month earlier than the House measure, which would stretch to mid-December.

Senate Democrats have said an earlier end date would give them more time to pass tailored spending bills for the rest of 2014 and reach a deal with Republicans to avoid the sequester cuts. Democrats hope to replace the sequester with a mix of more targeted cuts and additional revenue. Coming to terms on such a deal would be difficult, however.

Ending the short-term spending measure in mid-November would leave time for Congress to “get to work and enact 12 fiscally responsible appropriations bills, lay the groundwork for canceling sequester for two years, and invest in America’s needs today and the needs of the future,” Senate Appropriations Committee Chairwoman Barbara Mikulski (D., Md.) said on the Senate floor last week.

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